Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles

Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles

G. W. Hanson,^1,a)R. C. Monreal,²and S. P. Apell³

1Department of Electrical Engineering, University of Wisconsin-Milwaukee 3200 N. Cramer St., Milwaukee, Wisconsin 53211, USA

2Departamento de Fı´sica Teo´rica de la Materia Condensada C05, Universidad Auto´noma de Madrid, 28049 Madrid Spain

3Department of Applied Physics, Chalmers University of Technology, SE-41296 Go¨teborg, Sweden (Received 12 April 2011; accepted 12 May 2011; published online 22 June 2011)

We report on the absorption of electromagnetic radiation by metallic nanoparticles in the radio and far infrared frequency range, and subsequent heating of nanoparticle solutions. A recent series of papers has measured considerable radio frequency (RF) heating of gold nanoparticle solutions. In this work, we show that claims of RF heating by metallic nanoparticles are not supported by theory. We analyze several mechanisms by which nonmagnetic metallic nanoparticles can absorb low frequency radiation, including both classical and quantum effects. We conclude that none of these absorption mechanisms, nor any combination of them, can increase temperatures at the rates recently reported. A recent experiment supports this finding.V^C 2011 American Institute of Physics.

[doi:10.1063/1.3600222]

I. INTRODUCTION

Absorption of electromagnetic radiation by metallic nanoparticles in the visible and infrared parts of the energy spectrum has attracted renewed interest due to potential applications in medicine and biology.¹ The irradiated par- ticles can absorb energy, one channel being the excitation of surface plasmons, and this energy can also be transferred to their surrounding medium leading to a variety of uses, such as markers or in ablation of cancer tumors. Individual nano- particles must be subject to an intense electric field to heat substantially; e.g., in Ref. 2 individual gold nanoparticles heated by several hundred degrees when the incident laser intensity wasI 1 GW/m², yet individual nanoparticles are projected to heat far less than a degree for intensities on the order of 10⁴ W/m² (Ref. 3). However, even for the latter case, large collections of nanoparticles can generate signifi- cant heating (tens to hundreds of degrees) over macroscopic volumes.³Alternately, because of the relatively low penetra- tion depth of radiation at infrared (IR) and visible frequen- cies, radio frequency (RF) energy has been suggested for biomedical applications because it penetrates easily in the human body, thus reaching important internal organs. A se- ries of papers^4–12 has considered RF heating of gold nano- particles, showing significant heating. However, a recent experiment¹³ contradicts this, and finds that gold nanopar- ticles at RF do not heat. It was concluded that at RF the observed heating was due to Joule heating of the ionic stock solution which housed the nanoparticles. The purpose of this paper is to show that absorption of MHz through low THz radiation by individual, noninteracting, spherical metallic nanoparticles is negligible and produces no substantial heat- ing and, further, that the heating is so insignificant that even very large collections of nanoparticles cannot lead to heating of the suspension, supporting the conclusions of Ref.13. To

this end, we analyze several mechanisms by which nonmag- netic metallic nanoparticles can absorb low frequency radia- tion, namely, (i) the classical prediction of Mie theory for the absorption of electromagnetic energy by small spheres, including the possibility that the spheres are coated by an absorbing material, and (ii) electronic absorption occurring at the surface of the spheres due to electron spillout, surface roughness, and effects of surface phonons. We conclude that none of these absorption mechanisms, nor any combination of them, can increase temperatures at the rates recently reported in Refs.4–12at MHz frequencies. We note that the same conclusions can be made for frequencies at least through the low THz regime.

II. CLASSICAL THEORY

For a general material sphere of radius R immersed in an infinite host medium having real-valued relative permit- tivity eh, the extinction and scattering cross sections are¹⁴

Cext¼2p k²_h

X¹

n¼1

2nþ 1

ð ÞRe að nþ bnÞ; (1)

Csca¼2p k²_h

X¹

n¼1

2nþ 1

ð Þ aj jn ²þ bj jn²

; (2)

wherekh¼ x=cð Þ ffiffiffiffieh

p , x is the angular frequency of the EM field,c is the speed of light in vacuum and anandbnare the Mie coefficients representing electric and magnetic multi- poles. The absorption cross section is Ca¼ Cext Csca. For very small material spheres (khR 1) we can ignore scatter- ing compared to extinction.

For a small nonmagnetic sphere, the leading terms of the Mie coefficients (e^ixtdependence) are

a1¼ i2ðkhRÞ³ 3

e eh

eþ 2eh

a)Electronic mail: [email protected]. (3)

0021-8979/2011/109(12)/124306/6/$30.00 109, 124306-1 VC2011 American Institute of Physics

b1¼ iðkhRÞ⁵ 45

e eh

eh

 

: (4)

The corresponding absorption cross section of the sphere is then

Ca’6p

k_h²Re að 1þ b1Þ ¼ kh Im að ^eþ a^mÞ  C^e_aþ C^m_a where a^e is the electric dipole polarizability and a^m is the magnetic dipole polarizability (associated with eddy cur- rents). Since we are studying good conductors at low fre- quencies, b1 can dominate a1 despite the extra factor of

khR

ð Þ², except for the very smallest spheres.

We will assume eðxÞ to be given by the Drude formula e xð Þ ¼ 1  x²_p

x xð þ i=s0Þ; (5) where s₀ is the bulk relaxation time and x_p is an effective plasma frequency. This is a good fit for noble metals at low frequencies.

A. Small coated sphere

In many instances nanoparticles may have a coating, such as an oxide layer, sodium citrate used in gold nanopar- ticle fabrication, or a protein for selective binding to a cell.

For a sphere having radiusR and relative permittivity e, with a coating having thickness d and relative permittivity ecoat, embedded in a host medium having real-valued relative per- mittivity eh, the electric dipole polarizability is¹⁴

a^e;cs¼ 3Vcnp

 ðecoat ehÞ e þ 2eð coatÞ þ rcðe ecoatÞ eðhþ 2ecoatÞ ecoatþ 2eh

ð Þ e þ 2eð coatÞ þ 2rcðecoat ehÞ e  eð coatÞ; (6) where Vcnp is the volume of the coated particle and rc R= R þ d½ ð Þ³. The magnetic polarizability is

a^m;cs¼Vcnp

10 ðkhRÞ² e ecoat

eh

 

1þ½khðRþ dÞ² 3

( )

(7)

which is very weakly dependent on coating thickness. Even if the core is a good metal, the presence of an absorptive coating causes the electric dipole contribution to be very large compared to the magnetic dipole contribution through most of the GHz range.

The presence of even a very thin, slightly absorbing coat- ing can have a profound influence on the absorbing properties of a particle.^15–17For example, let the host medium be an in- sulator with e_h¼ 80, and assume an R ¼ 20 nm gold sphere (x_p’ 1:37  10¹⁶ s^1, s₀’ 2  10^14s, which corresponds to a dc conductivity of r¼ 3:3  10⁷ S/m). The absorption cross section using s₀, normalized by the cross-sectional area A¼ pR², is shown in Fig. 1(solid line) from 1 MHz to 1 THz. The addition of a 2 nm thick coating withReðecoatÞ ¼ 4 and r_coat¼ 0.01 S/m [Im eðcoatÞ ¼ 13.8 at 13 MHz] increases

the absorption cross section by six orders of magnitude com- pared to the uncoated sphere at a typical RF frequency of 13 MHz, as seen in Fig.1(dashed line – in this case the absorp- tion cross section is normalized by A¼ p R þ dð Þ²’ pR²).

Also shown in Fig.1are the corresponding values computed using an effective relaxation time s_eff accounting for nonlocal effects, described in Sec.III. Note that the host medium hav- ing a constant value of e_h ¼ 80 can be considered to be pure water up until approximately 1 GHz, after which water disper- sion and absorption become important; these effects will be considered later.

In Fig.1, we have assumed a mildly conducting coating;

Eq. (40)in Ref. 17gives the coating conductivity that maxi- mizes absorption, and the value r¼ 0:01 S/m maximizes absorption in a wide region of frequencies starting near 10 MHz. Many practical coatings may be expected to have con- ductivities on the order of 0:001 1 S/m; one estimate¹⁸ of the conductivity of sodium citrate buffer solutions is 0:25 S/m.

However, as we will show below, even for the coated sphere the absorption cross section is approximately eight orders of magnitude too small to account for the heating observed in the aforementioned papers (e.g., Ref. 7), even with a coating that maximizes absorption, and, in fact, this is true through the low THz regime. A model has been pro- posed in Ref.7to explain the observed RF heating, but this model cannot be correct because it computes absorption by multiplying conductivity inside the particle by the external (applied) electric field value, rather than using the electric field inside the particle (which, due to the high conductivity of the particle, is smaller by many orders of magnitude com- pared to the applied field).

The inability to explain significant RF absorption using the classical model leads to consideration of other factors, such as nonlocal surface effects. These have earlier been shown to increase the absorption cross section of small spheres in the visible or IR part of the EM spectrum.^19,20 Their role at RF through THz frequencies is investigated next.

FIG. 1. (Color online) Absorption cross section normalized by cross-sec- tional areaA¼ pR²for aR¼ 20 nm gold sphere immersed in an insulating host medium having e_h¼ 80 (solid line, s ¼ s₀). The result for a coated sphere havingd¼ 2 nm, Reecoat¼ 4, and rcoat¼ 0:01 S/m is also shown [dashed line,A¼ p R þ dð Þ²]. The curves for s¼ seff account for nonlocal effects as discussed in Sec.III.

III. NONLOCAL ELECTRONIC SURFACE EFFECTS – ELECTRON SPILLOUT AND SURFACE ROUGHNESS

It has been known for a long time^20,21that nonlocal sur- face effects on the electromagnetic response of small spheres (a few nanometers in size) can be described by means of two lengths,dranddh, defined as

drðxÞ ¼ 1

e eh 1

ð1 0

drErðrÞ  E^cl_rðrÞ

E^cl_rðRÞ (8) and

dhðxÞ ¼ ð1

0

drr R

DhðrÞ  D^cl_hðrÞ

E^cl_hðRÞ : (9) In Eqs.(8)and(9),Eris the radial component of the electric field and Dh is the component of the electric displacement tangential to the surface for a realistic (non-abrupt) surface model;E^cl_r andD^cl_h are their classical counterparts for an ab- rupt particle boundary. Since the actual EM fields are contin- uous functions at the metal/medium interface, contrarily to the classical theory,dranddhmeasure the response of a real- istic metal nanoparticle surface to external radiation. In par- ticular, the imaginary parts of both functions are related to absorption of energy by surface modifications of the classical bulk response. While Imdr describes absorption by electron- hole pairs taking into account the actual electron density pro- file at the surface – electron spillout, Imdh describes absorp- tion by diffuse electron surface scattering. If we assume a classical abrupt interface then both dr¼ 0 and dh¼ 0 by their definition in Eqs.(8)and(9), respectively. For realistic surfaces,dr;hare typically in the range of 0:1 nm.

These surface effects can be appropriately taken into account by defining an effective dielectric function for a metal nanoparticle as^20,22

eeffðxÞ ¼ e

1þ2 e

dh

R 1þ ð_e^e

h 1Þ^d_R^r: (10)

Then, the electric dipolar contribution to the absorption cross section is

C^e_a¼x

cVnpe³⁼²_h 3e eþ 2eh









2

Im1 eeff

; (11)

whereVnp¼ 4pR³=3 is the nanoparticle volume and where all permittivities are relative. Thedrcorrection is not present for the magnetic dipole contribution since that is governed by the tangential electric fields, and it has been shown that for the magnetic dipole contribution the effect ofdh is very small,²³ so that we just apply the surface corrections to the electric dipole contribution.

Sincedh=eR is small, we can make the expansion

1 eeff

’1 e þ 1

e1 eh

 

dr

R þ2 e²

dh

R: (12)

The first term in this equation gives the classical volume con- tribution to the absorption cross section (i.e.,C^e_a in Sec. II),

the second gives the electronic spill-out surface contribution and the third gives the surface contribution due to electronic surface scattering. Since we are interested in the low fre- quency regime the particle conductivity is very large and the uniform interior field is very small in the particle bulk. There- fore, we expect the surface contributions to be more important than the volume one.

Next, we estimate each contribution separately, which is a novel analysis for low frequencies. We take the limits x xp and xs₀ 1 in the bulk dielectric function. Then, the classical volume effect is

Im1 e ’ x

xp

1 xps0

: (13)

With respect to the surface contributions, we also approxi- mate dr and dh by their corresponding values for a planar surface. This is justified because these lengths are of the order of a few A˚ ngstro¨m while particle radii are much larger than this. The surface contribution in Eq.(12)due to excita- tion of surface electron-hole pairs is^20,24

Im 1 e1

eh

 

dr

R



’ 1 eh

Imdr

R ’ 1

6e_h

 hx EF

gðrsÞ 1

kFR (14) where EF andkF are the Fermi energy and the Fermi wave vector, respectively, and g is a number depending on the metal via its one-electron radiusrs. g has been estimated to be of the order of 1 in Ref.24.

Finally, in order to estimate the surface contribution to the absorption cross section due to electronic surface scatter- ing, we will adopt the simple model of Ref.22in which the electron scattering is determined by a phenomenological roughness parameterp, 0 p  1. A very rough inhomoge- neous surface is described by p¼ 0, while p ¼ 1 for a per- fectly reflecting surface. The use of the semiclassical infinite barrier model for the metal/medium interface also allows for a simple analytical evaluation ofdh, as expressed by Eq.(12) in Ref.22. In the low frequency limit, this yields,

Im 2 e²

dh

R

 

’1 p ffiffiffiffiffi

p10ðxs0Þ¹⁼² xps0

vF

xpR: (15) Note that the absorption cross section goes to zero as x goes to zero. However, the contribution due to the microscopic surface roughness to 1=eeff shows a slower decrease with x, going as x¹⁼²instead of x.

Putting all the contributions together, we can finally write the effective dielectric function as

eeff ’ ix²_pseff

x (16)

where 1=s_effis given by 1

seff

¼ 1 s0

þ 1 6eh

 hxp

EF

gðrsÞ xp

kFRþ1 p ffiffiffiffiffi p10 1

ðxs0Þ¹⁼² vF

R: (17) We can use this effective dielectric function to calculate the electric dipolar part of the absorption cross section for coated

or uncoated spheres taking into account surface effects. In writing Eq. (16), we have taken into account that the real part of e_eff is going to be unimportant since its imaginary part will dominate at small frequencies. Note that the last term in Eq.(17)is x-dependent and can dominate the value of s_eff at sufficiently low frequencies, independent of particle size.

To reconsider the previous example of a gold R¼ 20 nm sphere in water atf ¼ 13 MHz, using p ¼ 0 we obtain seff ¼ 5:8  10^17 s, a more than two order of magnitude decrease from the bulk value (most of which is contributed bydh; forp¼ 1, seff ¼ 1:5  10^14s). The resulting absorp- tion cross section is dominated by the electric dipole contri- bution²⁵ and at 13 MHz exhibits an increase of nearly two orders of magnitude compared to the values obtained using s0, as shown in Fig.1(solid line indicated by s_eff). However, if we add the same coating as above, the absorption cross section is unaffected by the value of the relaxation time (dashed line indicated by s_eff) since most absorption occurs in the coating.

Changes in s due to grain boundary scattering may also be relevant for bare nanoparticles, but, again, an absorptive coating (or absorptive host) will render any such changes to s unimportant. Another possible mechanism is enhanced absorption through the excitation of surface phonons. How- ever, as shown in Ref²⁶even at 1.7 THz the phonon contri- bution is smaller than the electron-hole pair surface contribution, and the phonon contribution decays rapidly with lower frequencies. Finally, we want to mention that there have been several attempts in the literature to include surface roughness in the optical response, see e.g. Ref.27 and²⁸ However, since we have no clear information about the surface morphology, their relevance for the present work cannot be clearly judged.

IV. THERMODYNAMICAL ANALYSIS

In the following we show that absorption by non-inter- acting spherical metal nanoparticles cannot cause the heating observed in the reported experiments. To do this, we solve the equation for the flow of heat in a medium characterized by its thermal conductivity, specific heatcv, and mass den- sity q. We assume that the heat source is a collection ofNp

point sources (representing the nanoparticles) placed at posi- tionsRp. The heat equation is

jr²Tðr; tÞ  qcv

@T

@tþ Qðr; tÞ ¼ 0; (18) the expression for the sourceQ (W/m³) being

Qðr; tÞ ¼ hðtÞQ0

X

Rp

dðr  RpÞ; (19)

where hðtÞ is the Heaviside function. Here, Q0 is the energy absorbed per unit time by one nanoparticle. We are implic- itly assuming that the absorbed EM energy is efficiently con- verted to heat.²⁹Equations. (18)and(19) can be solved by standard methods considering an infinite medium,³⁰ giving the rate of increase in temperature as

dTðr; tÞ dt ¼ Q0

ffiffiffiffiffiffiffiffiffiffi pqcv

p 8p²ðjtÞ³⁼²

X

R_p

exp jr  Rpj² 4_qc^jt

v

" #

: (20)

From Eq.(20)the temperatureTðr; tÞ is obtained by integra- tion. The temperature thus obtained is consistent with Eq.(2) of Ref.3.

Since we have Np terms in the sum appearing in Eq.

(20), we divide this sum byNp and, correspondingly, multi- ply Q0 by Np. Then we compute dTðr; tÞ=dt for a constant value ofNpQ0¼ 1 W. This allows us to plot results for dif- ferent numbers of particles on the same scale. For simplicity, we choose a system with cubic symmetry and put theNppar- ticles in a cubic volumeL³. In Fig.2,L¼ 1 cm (1 mL vol- ume), corresponding to the approximate volume of the experimental cuvette in Ref.7. We use parameters appropri- ate for water: cv¼ 4:18  10³ J/kg-K, q¼ 1000 kg/m³, j¼ 0:61 W/m-K.

Results are shown for Np ¼ 100³ (solid lines) and Np¼ 500³ (dashed lines) particles and at different points r¼ ðx; y; zÞ, x ¼ y ¼ z, nm (top curve), x ¼ 1 mm (middle curve), andx¼ 2:5 mm (lower curve) with the origin placed at the center of the cubic cuvette. We see that the temperature is rather uniform over the cuvette and is essentially independ- ent of the number of particles. Note also that one needs NpQ0 1 W to obtain significant heating. It is easy to see that this level of heating is impossible to achieve for metal nanoparticles at RF-THz frequencies. For example, assume the same nanoparticles as in the previous examples (gold, R¼ 20 nm) at 13 MHz. Consider a spherical cuvette of radius a¼ 7:1 mm (1:5 mL volume) in air and containing a deion- ized water and nanoparticle solution. Assume a field of 15 kV/m in air, the maximum field considered in, e.g., Ref. 7.

The electric field inside the cuvette is 550 V/m, and the inten- sity incident on each nanoparticle is I¼ 3:6  10³ W/m². Then, Q^bare₀ ¼ C^bare_a I’ 1:6  10^24 W and Q^coated₀ ¼ C^coated_a I’ 1  10^19 W. Even assuming a volume fraction of one (Np  10¹⁶) we would have for the coated sphere NpQ^coated₀  0:001 W, which is insufficient for heating. For

FIG. 2. (Color online) Temperature vs time at different pointsr¼ x; y; zð Þ, x ¼ y ¼ z, with the origin at the center of a cube of side L ¼ 1 cm for Np¼ 100³ (solid lines) and Np¼ 500³ (dashed lines). x¼ 10 nm (top curve),x¼ 1 mm (middle curve), and x ¼ 2:5 mm (lower curve).

the small volume fractions used in the RF experiments ( 10^6), or for any other reasonable volume fraction that may be considered, no RF heating should be observed, con- sistent with the findings of Ref.13. The same can be said for GHz frequencies. At, e.g., 1 GHz, C^bare_a I’ 1:3  10^21 W andC^coated_a I’ 1  10^19W.

Further conclusions can be drawn from the steady state analysis. Assuming non-interacting, non-aggregated nano- particles, the steady state temperature at the surface of a spherical cuvette is¹⁷

DT¼ NpQ0

4pajm

(21) wherea is the radius of the cuvette and jmis the thermal con- ductivity of the medium outside of the cuvette. The associated heating rate (degrees/s) is,HR¼ NpQ0qcvV, where V is the volume of the cuvette. Assuming a 1:5 mL volume and thermal conductivity same as water, the necessary power (W) to increase the temperature DT is NpQ0 ¼ 4pajmDT¼ 0:055DT.

For DT¼ 20 degrees as in Fig. 2, NpQ0¼ 1:1 W, in good agreement with the transient analysis. The heating rate is HR¼ 0:16 /s, also in good agreement with the linear part of the heating curves. If the cuvette resides in air, the low thermal conductivity of the air results in NpQ0 ¼ 0:05 W needed to reach DT¼ 20 degrees. Expressed in terms of volume fraction F _v, the required value of CaI to yield a DT temperature increase is, according to Eq.(21),

CaI¼4pj_mR³DT

Fva² : (22)

ForR¼ 20 nm and considering a 1:5 mL spherical cuvette in air, for a volume fraction of 10^6 we would need CaI order 10^12W (i.e., for each nanoparticle) to achieve DT in the tens of degrees. Assuming electric fields in the kV/m range outside the cuvette, this order of absorption cross sec- tion cannot be achieved at RF, where CaI is 10^19 W per nanoparticle in the example above.

The conclusions above pertain to nanoparticles in a non- absorbing host medium. It is worthwhile to see if nanopar- ticles can contribute to heating in an absorbing host medium, or if the host medium absorption itself is the dominant con- tribution. In Ref. 13, the observed RF heating was com- pletely attributed to Joule heating of the ionic host medium.

The authors also argue that this mechanism is likely respon- sible for the heating observed in Refs.31–33 in the 1 12 GHz range.

V. ABSORBING HOST MEDIUM

To investigate an absorbing host medium we consider nanoparticles in liquids characterized by the Debye (Cole- Cole) form

erð Þ ¼ ex rð1Þ þerð Þ  e0 rð Þ1 1 ixsð wÞ^1aþ i r

xe0

: (23)

We consider pure water (experimentally, this would be de- ionized water) at 25 C, where e_rð Þ ¼ 78:36, e0 rð1Þ ¼ 5:2,

a¼ 0:02, r ¼ 0, and sw¼ 8:27 ps (Ref. 34) (1=2ps_w¼ 19 GHz), and physiological saline, where erð Þ ¼ 76,0 erð1Þ ¼ 4:5, a ’ 0.02, r ¼ 1.45 S/m, and s ¼ 8:11 ps.

Since Mie theory is only applicable to a nonabsorbing host medium, we compute the effective permittivity e_eff of the nanoparticle—liquid solution using the Maxwell-Garnett mixing formula.³⁵ From this, we compute the absorption cross section of a 1:5 mL cuvette containing pure water or saline solution in air. Figure 3shows the absorption cross section normalized by cross-sectional area of the cuvette (A¼ 1:58  10^4 m²) for a moderately-large nanoparticle volume fraction of 0:01, for three different values of nano- particle conductivity, r_part¼ 0, 0:1 and 10⁷ S/m, and for unity nanoparticle permittivity. For the saline host medium, absorption is clearly independent of the presence or absence of the nanoparticles, regardless of nanoparticle conductivity (the three curves are indistinguishable on the scale of the plot). This indicates that spherical nanoparticles cannot enhance absorption in saline-like fluids in the considered fre- quency range. For the water host, below approximately 100 MHz water absorption is small enough so that nanoparticles with an appropriate value of conductivity can enhance absorption. From Ref. 17, r_part¼ 0:1 S/m approximately maximizes absorption in the low MHz range, hence the enhanced absorption shown by the long-dashed curve in Fig.

3. At these lower frequencies the absorption of metallic nanoparticles (r¼ 10⁷ S/m) is too small, even at the high volume fraction considered, to enhance absorption compared to that of the water. In fact, replacing the metallic nanopar- ticles with air voids having r¼ 0 S/m (solid line) leads to the same net absorption. Above approximately 100 MHz the absorption of the nanoparticle-water solution is independent of the presence or absence of nanoparticles of any material composition. Therefore, in all cases involving metal nano- particles, absorption and subsequent heating is due to the absorbing host medium, and is not influenced by the pres- ence or absence of spherical nanoparticles. However, needle- like objects such as carbon nanotubes or elongated ellipsoi- dal particles maintain a strong radial near-field in the vicinity

FIG. 3. (Color online) Absorption cross section of a 1:5 mL spherical cuv- ette in air, normalized by cross-sectional area of the cuvette, for a nanopar- ticle volume fraction of 0:01 and for three different values of nanoparticle conductivity (r¼ 0; 0:1 and 10⁷S/m).

of the tube ends (e.g., lightning-rod effect) that can signifi- cantly enhance absorption in a conducting host.³⁶

Finally, it should be noted that nanoparticle agglomera- tion has not been considered here, and could possibly con- tribute to increased heating.

In conclusion, we have analyzed several mechanisms by which spherical nanospheres can absorb RF–far infrared radiation which can be subsequently released to their host medium as heat. Considering volume and surface effects we determine absorption cross sections. By performing a ther- modynamical analysis we find that these values of cross sec- tions at the incident power levels typical of experiments are unable to produce the increase in temperature of tens of degrees reported in many RF-GHz experiments. Our analysis also confirms recent experimental results showing that the Joule heating of the solution containing the nanoparticles is the mechanism producing such rises of temperatures, and that the presence of nonaggregated metallic nanoparticles plays no role in heating. We confirm that Joule heating of the host medium (including pure water) should also be the heat- ing mechanism operating in the microwave range of the EM spectrum.

ACKNOWLEDGMENTS

Support by the Spanish Ministerio de Ciencia e Innova- cio´n Grant No. FIS2008-04209 and the Swedish Foundation for Strategic Research (metamaterial Grant No. SSF RMA08-0109) is acknowledged.

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Appl. Phys.108, 114302 (2010).